If "your religion requires cognitive dissonance!" were a persuasive argument against religion, no one would bother talking about religion.

Less cynically, your argument does not lead to the contradiction you want it to. Two main reasons. First, that's not the reasoning the Catholic Church uses. They go old-school - Leviticus 20:13 reads:

If a man lies with a male as he lies with a woman, both of them have committed an abomination. Homosexuality being immoral is thus not a conclusion so much as a premise. God doesn't get much clearer than calling something an "abomination."

Your argument arguably misses the point by attacking "natural" birth control. The biblical proscription against birth control comes from the (mis)interpretation of the story of Onan, where God kills a man for spilling his seed instead of impregnating his dead brother's wife. In the "rhythm" method, you're not using any artificial intervention to make it such that seed is spilled, and a significant risk of pregnancy in fact remains. By contrast, birth control and condoms cause barrenness and spilling respectively, so they aren't OK. Yes, this argument is making a rather arbitrary distinction, but unless you can squelch that distinction, you don't get the logical necessity you need. A clearer counterargument would be allowing postmenopausal women to marry, or even married post-menopausal women to have sex. Again, though, it's natural; they aren't failing to get pregnant due to their own actions, they're failing because they just can't do it, and if the church wants to claim that as the distinction, it has a consistent system. Stupid and baseless, but consistent.